Risk, Return, and Volatility Feedback: A Bayesian Nonparametric Analysis

The relationship between risk and return is one of the most studied topics in finance. The majority of the literature is based on a linear, parametric relationship between expected returns and conditional volatility. However, there is no theoretical justification for the relationship to be linear. This paper models the contemporaneous relationship between market excess returns and log-realized variances nonparametrically with an infinite mixture representation of their joint distribution. With this nonparametric representation, the conditional distribution of excess returns given log-realized variance will also have a infinite mixture representation but with probabilities and arguments depending on the value of realized variance. Our nonparametric approach allows for deviation from Gaussianity by allowing for higher order non-zero moments. It also allows for a smooth nonlinear relationship between the conditional mean of excess returns and log-realized variance. Parsimony of our nonparametric approach is guaranteed by the almost surely discrete Dirichlet process prior used for the mixture weights and arguments. We find strong robust evidence of volatility feedback in monthly data. Once volatility feedback is accounted for, there is an unambiguous positive relationship between expected excess returns and expected log-realized variance. This relationship is nonlinear. Volatility feedback impacts the whole distribution and not just the conditional mean.